The appeal came at a time of growing political unrest in the Arab world. Popular uprisings have deposed the leaders of Tunisia and Egypt, and anti-government protests are gaining momentum in Libya, Yemen and Bahrain.
In the 10-minute recording, Al-Rubeish criticized Saudi Arabia for providing a haven for deposed Tunisian President Zine El Abidine Ben Ali.
He also said toppling longtime rulers is not enough and that new governments must be established based on Islamic religious law, or Shariah.
"One tyrant goes, only to be replaced another who may fix for the people some of their worldly issues by offering job opportunities and increasing their income, but the greater problem remains," al-Rubeish said, according to a translation provided by SITE.